Number Theory
Naoki Sato <sato@artofproblemsolving.com>
0
Preface
This set of notes on number theory was originally written in 1995 for students
at the IMO level. It covers the basic background material that an IMO
student should be familiar with. This text is meant to be a reference, and
not a replacement but rather a supplement to a number theory textbook;
several are given at the back. Proofs are given when appropriate, or when
they illustrate some insight or important idea. The problems are culled from
various sources, many from actual contests and olympiads, and in general
are very difficult. The author welcomes any corrections or suggestions.
1
Divisibility
For integers a and b, we say that a divides b, or that a is a divisor (or
factor) of b, or that b is a multiple of a, if there exists an integer c such
that b = ca, and we denote this by a | b. Otherwise, a does not divide b, and
we denote this by a - b. A positive integer p is a prime if the only divisors of
p are 1 and p. If pk | a and pk+1 - a where p is a prime, i.e. pk is the highest
power of p dividing a, then we denote this by pk ka.
Useful Facts
• If a, b > 0, and a | b, then a ≤ b.
• If a | b1 , a | b2 , . . . , a | bn , then for any integers c1 , c2 , . . . , cn ,
a|
n
X
bi c i .
i=1
Theorem 1.1. The Division Algorithm. For any positive integer a and
integer b, there exist unique integers q and r such that b = qa + r and
0 ≤ r < a, with r = 0 iff a | b.
1
Theorem 1.2. The Fundamental Theorem of Arithmetic. Every integer
greater than 1 can be written uniquely in the form
pe11 pe22 · · · pekk ,
where the pi are distinct primes and the ei are positive integers.
Theorem 1.3. (Euclid) There exist an infinite number of primes.
Proof. Suppose that there are a finite number of primes, say p1 , p2 , . . . ,
pn . Let N = p1 p2 · · · pn + 1. By the fundamental theorem of arithmetic, N
is divisible by some prime p. This prime p must be among the pi , since by
assumption these are all the primes, but N is seen not to be divisible by any
of the pi , contradiction.
Example 1.1. Let x and y be integers. Prove that 2x + 3y is divisible
by 17 iff 9x + 5y is divisible by 17.
Solution. 17 | (2x + 3y) ⇒ 17 | [13(2x + 3y)], or 17 | (26x + 39y) ⇒
17 | (9x + 5y), and conversely, 17 | (9x + 5y) ⇒ 17 | [4(9x + 5y)], or
17 | (36x + 20y) ⇒ 17 | (2x + 3y).
Example 1.2. Find all positive integers d such that d divides both n2 +1
and (n + 1)2 + 1 for some integer n.
Solution. Let d | (n2 + 1) and d | [(n + 1)2 + 1], or d | (n2 + 2n + 2).
Then d | [(n2 + 2n + 2) − (n2 + 1)], or d | (2n + 1) ⇒ d | (4n2 + 4n + 1), so
d | [4(n2 +2n+2)−(4n2 +4n+1)], or d | (4n+7). Then d | [(4n+7)−2(2n+1)],
or d | 5, so d can only be 1 or 5. Taking n = 2 shows that both of these
values are achieved.
Example 1.3. Suppose that a1 , a2 , . . . , a2n are distinct integers such
that the equation
(x − a1 )(x − a2 ) · · · (x − a2n ) − (−1)n (n!)2 = 0
has an integer solution r. Show that
a1 + a2 + · · · + a2n
.
r=
2n
(1984 IMO Short List)
Solution. Clearly, r 6= ai for all i, and the r − ai are 2n distinct integers,
so
|(r − a1 )(r − a2 ) · · · (r − a2n )| ≥ |(1)(2) · · · (n)(−1)(−2) · · · (−n)| = (n!)2 ,
2
with equality iff
{r − a1 , r − a2 , . . . , r − a2n } = {1, 2, . . . , n, −1, −2, . . . , −n}.
Therefore, this must be the case, so
(r − a1 ) + (r − a2 ) + · · · + (r − a2n )
= 2nr − (a1 + a2 + · · · + a2n )
= 1 + 2 + · · · + n + (−1) + (−2) + · · · + (−n) = 0
a1 + a2 + · · · + a2n
.
⇒ r=
2n
Example 1.4. Let 0 < a1 < a2 < · · · < amn+1 be mn + 1 integers. Prove
that you can select either m + 1 of them no one of which divides any other,
or n + 1 of them each dividing the following one.
(1966 Putnam Mathematical Competition)
Solution. For each i, 1 ≤ i ≤ mn + 1, let ni be the length of the longest
sequence starting with ai and each dividing the following one, among the
integers ai , ai+1 , . . . , amn+1 . If some ni is greater than n then the problem
is solved. Otherwise, by the pigeonhole principle, there are at least m + 1
values of ni that are equal. Then, the integers ai corresponding to these ni
cannot divide each other.
Useful Facts
• Bertrand’s Postulate. For every positive integer n, there exists a prime
p such that n ≤ p ≤ 2n.
• Gauss’s Lemma. If a polynomial with integer coefficients factors into
two polynomials with rational coefficients, then it factors into two polynomials with integer coefficients.
Problems
1. Let a and b be positive integers such that a | b2 , b2 | a3 , a3 | b4 , b4 | a5 ,
. . . . Prove that a = b.
2. Let a, b, and c denote three distinct integers, and let P denote a polynomial having all integral coefficients. Show that it is impossible that
P (a) = b, P (b) = c, and P (c) = a.
(1974 USAMO)
3
3. Show that if a and b are positive integers, then
n n
1
1
a+
+ b+
2
2
is an integer for only finitely many positive integers n.
(A Problem Seminar, D.J. Newman)
4. For a positive integer n, let r(n) denote the sum of the remainders when
n is divided by 1, 2, . . . , n respectively. Prove that r(k) = r(k − 1) for
infinitely many positive integers k.
(1981 Kürschák Competition)
5. Prove that for all positive integers n,
0<
n
X
g(k)
k
k=1
−
2n
2
< ,
3
3
where g(k) denotes the greatest odd divisor of k.
(1973 Austrian Mathematics Olympiad)
6. Let d be a positive integer, and let S be the set of all positive integers
of the form x2 + dy 2 , where x and y are non-negative integers.
(a) Prove that if a ∈ S and b ∈ S, then ab ∈ S.
(b) Prove that if a ∈ S and p ∈ S, such that p is a prime and p | a,
then a/p ∈ S.
(c) Assume that the equation x2 + dy 2 = p has a solution in nonnegative integers x and y, where p is a given prime. Show that if
d ≥ 2, then the solution is unique, and if d = 1, then there are
exactly two solutions.
2
GCD and LCM
The greatest common divisor of two positive integers a and b is the greatest positive integer that divides both a and b, which we denote by gcd(a, b),
and similarly, the lowest common multiple of a and b is the least positive
4
integer that is a multiple of both a and b, which we denote by lcm(a, b). We
say that a and b are relatively prime if gcd(a, b) = 1. For integers a1 , a2 ,
. . . , an , gcd(a1 , a2 , . . . , an ) is the greatest positive integer that divides all of
a1 , a2 , . . . , an , and lcm(a1 , a2 , . . . , an ) is defined similarly.
Useful Facts
• For all a, b, gcd(a, b) · lcm(a, b) = ab.
• For all a, b, and m, gcd(ma, mb) = m gcd(a, b) and lcm(ma, mb) =
mlcm(a, b).
• If d | gcd(a, b), then
gcd
a b
,
d d
=
gcd(a, b)
.
d
In particular, if d = gcd(a, b), then gcd(a/d, b/d) = 1; that is, a/d and
b/d are relatively prime.
• If a | bc and gcd(a, c) = 1, then a | b.
• For positive integers a and b, if d is a positive integer such that d | a,
d | b, and for any d0 , d0 | a and d0 | b implies that d0 | d, then d =
gcd(a, b). This is merely the assertion that any common divisor of a
and b divides gcd(a, b).
• If a1 a2 · · · an is a perfect k th power and the ai are pairwise relatively
prime, then each ai is a perfect k th power.
• Any two consecutive integers are relatively prime.
Example 2.1. Show that for any positive integer N , there exists a
multiple of N that consists only of 1s and 0s. Furthermore, show that if N
is relatively prime to 10, then there exists a multiple that consists only of 1s.
Solution. Consider the N + 1 integers 1, 11, 111, . . . , 111...1 (N + 1 1s).
When divided by N , they leave N + 1 remainders. By the pigeonhole principle, two of these remainders are equal, so the difference in the corresponding
integers, an integer of the form 111...000, is divisible by N . If N is relatively
prime to 10, then we may divide out all powers of 10, to obtain an integer of
the form 111...1 that remains divisible by N .
5
Theorem 2.1. For any positive integers a and b, there exist integers x
and y such that ax + by = gcd(a, b). Furthermore, as x and y vary over all
integers, ax + by attains all multiples and only multiples of gcd(a, b).
Proof. Let S be the set of all integers of the form ax+by, and let d be the
least positive element of S. By the division algorithm, there exist integers q
and r such that a = qd + r, 0 ≤ r < d. Then r = a − qd = a − q(ax + by) =
(1 − qx)a − (qy)b, so r is also in S. But r < d, so r = 0 ⇒ d | a, and
similarly, d | b, so d | gcd(a, b). However, gcd(a, b) divides all elements of S,
so in particular gcd(a, b) | d ⇒ d = gcd(a, b). The second part of the theorem
follows.
Corollary 2.2. The positive integers a and b are relatively prime iff there
exist integers x and y such that ax + by = 1.
Corollary 2.3. For any positive integers a1 , a2 , . . . , an , there exist
integers x1 , x2 , . . . , xn , such that a1 x1 +a2 x2 +· · ·+an xn = gcd(a1 , a2 , . . . , an ).
Corollary 2.4. Let a and b be positive integers, and let n be an integer.
Then the equation
ax + by = n
has a solution in integers x and y iff gcd(a, b) | n. If this is the case, then all
solutions are of the form
a
b
,
(x, y) = x0 + t · , y0 − t ·
d
d
where d = gcd(a, b), (x0 , y0 ) is a specific solution of ax + by = n, and t is an
integer.
Proof. The first part follows from Theorem 2.1. For the second part, as
stated, let d = gcd(a, b), and let (x0 , y0 ) be a specific solution of ax + by = n,
so that ax0 + by0 = n. If ax + by = n, then ax + by − ax0 − by0 = a(x − x0 ) +
b(y − y0 ) = 0, or a(x − x0 ) = b(y0 − y), and hence
(x − x0 ) ·
b
a
= (y0 − y) · .
d
d
Since a/d and b/d are relatively prime, b/d must divide x − x0 , and a/d must
divide y0 − y. Let x − x0 = tb/d and y0 − y = ta/d. This gives the solutions
described above.
6
Example 2.2. Prove that the fraction
21n + 4
14n + 3
is irreducible for every positive integer n. (1959 IMO)
Solution. For all n, 3(14n + 3) − 2(21n + 4) = 1, so the numerator and
denominator are relatively prime.
n
Example 2.3. For all positive integers n, let Tn = 22 + 1. Show that if
m 6= n, then Tm and Tn are relatively prime.
Solution. We have that
n
n−1
Tn − 2 = 22 − 1 = 22 ·2 − 1
2
= (Tn−1 − 1)2 − 1 = Tn−1
− 2Tn−1
= Tn−1 (Tn−1 − 2)
= Tn−1 Tn−2 (Tn−2 − 2)
= ···
= Tn−1 Tn−2 · · · T1 T0 (T0 − 2)
= Tn−1 Tn−2 · · · T1 T0 ,
for all n. Therefore, any common divisor of Tm and Tn must divide 2. But
each Tn is odd, so Tm and Tn are relatively prime.
Remark. It immediately follows from this result that there are an infinite
number of primes.
The Euclidean Algorithm. By recursive use of the division algorithm, we
may find the gcd of two positive integers a and b without factoring either,
and the x and y in Theorem 2.1 (and so, a specific solution in Corollary 2.4).
For example, for a = 329 and b = 182, we compute
329 = 1 · 182 + 147,
182 = 1 · 147 + 35,
147 = 4 · 35 + 7,
35 = 5 · 7,
and stop when there is no remainder. The last dividend is the gcd, so in
our example, gcd(329,182) = 7. Now, working through the above equations
7
backwards,
7 = 147 − 4 · 35 = 147 − 4 · (182 − 1 · 147)
= 5 · 147 − 4 · 182 = 5 · (329 − 182) − 4 · 182
= 5 · 329 − 9 · 182.
Remark. The Euclidean algorithm also works for polynomials.
Example 2.4. Let n be a positive integer, and let S be a subset of n + 1
elements of the set {1, 2, . . . , 2n}. Show that
(a) There exist two elements of S that are relatively prime, and
(b) There exist two elements of S, one of which divides the other.
Solution. (a) There must be two elements of S that are consecutive, and
thus, relatively prime.
(b) Consider the greatest odd factor of each of the n + 1 elements in
S. Each is among the n odd integers 1, 3, . . . , 2n − 1. By the pigeonhole principle, two must have the same greatest odd factor, so they differ
(multiplication-wise) by a power of 2, and so one divides the other.
Example 2.5. The positive integers a1 , a2 , . . . , an are such that each is
less than 1000, and lcm(ai , aj ) > 1000 for all i, j, i 6= j. Show that
n
X
1
< 2.
a
i
i=1
(1951 Russian Mathematics Olympiad)
1000
Solution. If m+1
< a ≤ 1000
, then the m multiples a, 2a, . . . , ma do
m
not exceed 1000. Let k1 the number of ai in the interval ( 1000
, 1000], k2 in
2
1000 1000
( 3 , 2 ], etc. Then there are k1 + 2k2 + 3k3 + · · · integers, no greater
than 1000, that are multiples of at least one of the ai . But the multiples are
distinct, so
k1 + 2k2 + 3k3 + · · · < 1000
⇒ 2k1 + 3k2 + 4k3 + · · · = (k1 + 2k2 + 3k3 + · · · ) + (k1 + k2 + k3 + · · · )
< 1000 + n
< 2000.
8
Therefore,
n
X
2
3
4
1
≤ k1
+ k2
+ k3
+ ···
a
1000
1000
1000
i=1 i
2k1 + 3k2 + 4k3 + · · ·
1000
< 2.
=
Note: It can be shown that n ≤ 500 as follows: Consider the greatest
odd divisor of a1 , a2 , . . . , a1000 . Each must be distinct; otherwise, two differ,
multiplication-wise, by a power of 2, which means one divides the other,
contradiction. Also, there are only 500 odd numbers between 1 and 1000,
from which the result follows. It also then follows that
n
X
3
1
< .
a
2
i=1 i
Useful Facts
• Dirichlet’s Theorem. If a and b are relatively prime positive integers,
then the arithmetic sequence a, a + b, a + 2b, . . . , contains infinitely
many primes.
Problems
1. The symbols (a, b, . . . , g) and [a, b, . . . , g] denote the greatest common
divisor and lowest common multiple, respectively of the positive integers a, b, . . . , g. Prove that
[a, b, c]2
(a, b, c)2
=
.
[a, b][a, c][b, c]
(a, b)(a, c)(b, c)
(1972 USAMO)
2. Show that gcd(am − 1, an − 1) = agcd(m,n) − 1 for all positive integers
a > 1, m, n.
9
3. Let a, b, and c be positive integers. Show that
lcm(a, b, c) =
abc · gcd(a, b, c)
.
gcd(a, b) · gcd(a, c) · gcd(b, c)
Express gcd(a, b, c) in terms of abc, lcm(a, b, c), lcm(a, b), lcm(a, c), and
lcm(b, c). Generalize.
4. Let a, b be odd positive integers. Define the sequence (fn ) by putting
f1 = a, f2 = b, and by letting fn for n ≥ 3 be the greatest odd divisor
of fn−1 + fn−2 . Show that fn is constant for n sufficiently large and
determine the eventual value as a function of a and b.
(1993 USAMO)
5. Let n ≥ a1 > a2 > · · · > ak be positive integers such that lcm(ai , aj ) ≤
n for all i, j. Prove that iai ≤ n for i = 1, 2, . . . , k.
3
Arithmetic Functions
There are several important arithmetic functions, of which three are presented here. If the prime factorization of n > 1 is pe11 pe22 · · · pekk , then the
number of positive integers less than n, relatively prime to n, is
1
1
1
1−
··· 1 −
n
φ(n) = 1 −
p1
p2
pk
= pe11 −1 pe22 −1 · · · pekk −1 (p1 − 1)(p2 − 1) · · · (pk − 1),
the number of divisors of n is
τ (n) = (e1 + 1)(e2 + 1) · · · (ek + 1),
and the sum of the divisors of n is
σ(n) = (pe11 + pe11 −1 + · · · + 1)(pe22 + pe22 −1 + · · · + 1)
· · · (pekk + pkek −1 + · · · + 1)
e1 +1
e2 +1
ek +1
p1 − 1
p2 − 1
pk
−1
=
···
.
p1 − 1
p2 − 1
pk − 1
Also, φ(1), τ (1), and σ(1) are defined to be 1. We say that a function
f is multiplicative if f (mn) = f (m)f (n) for all relatively prime positive
10
integers m and n, and f (1) = 1 (otherwise, f (1) = 0, which implies that
f (n) = 0 for all n).
Theorem 3.1. The functions φ, τ , and σ are multiplicative.
Hence, by taking the prime factorization and evaluating at each prime
power, the formula above are found easily.
Example 3.1. Find the number of solutions in ordered pairs of positive
integers (x, y) of the equation
1
1 1
+ = ,
x y
n
where n is a positive integer.
Solution. From the given,
1 1
1
+ =
⇔ xy = nx + ny ⇔ (x − n)(y − n) = n2 .
x y
n
If n = 1, then we immediately deduce the unique solution (2,2). For
n ≥ 2, let n = pe11 pe22 · · · pekk be the prime factorization of n. Since x, y > n,
there is a 1-1 correspondence between the solutions in (x, y) and the factors
of n2 , so the number of solutions is
τ (n2 ) = (2e1 + 1)(2e2 + 1) · · · (2ek + 1).
Example 3.2. Let n be a positive integer. Prove that
X
φ(d) = n.
d|n
Solution. For a divisor d of n, let Sd be the set of all a, 1 ≤ a ≤ n, such
that gcd(a, n) = n/d. Then Sd consists of all elements of the form b · n/d,
where 0 ≤ b ≤ d, and gcd(b, d) = 1, so Sd contains φ(d) elements. Also, it is
clear that each integer between 1 and n belongs to a unique Sd . The result
then follows from summing over all divisors d of n.
Problems
1. Let n be a positive integer. Prove that
n
X
τ (k) =
k=1
n j k
X
n
k=1
11
k
.
2. Let n be a positive integer. Prove that

2
X
X
τ 3 (d) = 
τ (d) .
d|n
d|n
3. Prove that if σ(N ) = 2N + 1, then N is the square of an odd integer.
(1976 Putnam Mathematical Competition)
4
Modular Arithmetic
For a positive integer m and integers a and b, we say that a is congruent to
b modulo m if m | (a − b), and we denote this by a ≡ b modulo m, or more
commonly a ≡ b (mod m). Otherwise, a is not congruent to b modulo m,
and we denote this by a 6≡ b (mod m) (although this notation is not used
often). In the above notation, m is called the modulus, and we consider the
integers modulo m.
Theorem 4.1. If a ≡ b and c ≡ d (mod m), then a + c ≡ b + d (mod m)
and ac ≡ bd (mod m).
Proof. If a ≡ b and c ≡ d (mod m), then there exist integers k and l
such that a = b + km and c = d + lm. Hence, a + c = b + d + (k + l)m, so
a + c ≡ b + d (mod m). Also,
ac = bd + dkm + blm + klm2
= bd + (dk + bl + klm)m,
so ac ≡ bd (mod m).
Useful Facts
• For all integers n,
2
n ≡
0
1
(mod 4)
if n is even,
if n is odd.
• For all integers n,
 

 0 
 if n ≡ 0
2
4 (mod 8) if n ≡ 2
n ≡
 

1
if n ≡ 1
12
(mod 4),
(mod 4),
(mod 2).
• If f is a polynomial with integer coefficients and a ≡ b (mod m), then
f (a) ≡ f (b) (mod m).
• If f is a polynomial with integer coefficients of degree n, not identically
zero, and p is a prime, then the congruence
f (x) ≡ 0
(mod p)
has at most n solutions modulo p, counting multiplicity.
Example 4.1. Prove that the only solution in rational numbers of the
equation
x3 + 3y 3 + 9z 3 − 9xyz = 0
is x = y = z = 0.
(1983 Kürschák Competition)
Solution. Suppose that the equation has a solution in rationals, with
at least one non-zero variable. Since the equation is homogeneous, we may
obtain a solution in integers (x0 , y0 , z0 ) by multiplying the equation by the
cube of the lowest common multiple of the denominators. Taking the equation modulo 3, we obtain x30 ≡ 0 (mod 3). Therefore, x0 must be divisible
by 3, say x0 = 3x1 . Substituting,
27x31 + 3y03 + 9z03 − 27x1 y0 z0 = 0
⇒ y03 + 3z03 + 9x31 − 9x1 y0 z0 = 0.
Therefore, another solution is (y0 , z0 , x1 ). We may then apply this reduction
recursively, to obtain y0 = 3y1 , z0 = 3z1 , and another solution (x1 , y1 , z1 ).
Hence, we may divide powers of 3 out of our integer solution an arbitrary
number of times, contradiction.
Example 4.2. Does one of the first 108 + 1 Fibonacci numbers terminate
with 4 zeroes?
Solution. The answer is yes. Consider the sequence of pairs (Fk , Fk+1 )
modulo 104 . Since there are only a finite number of different possible pairs
(108 to be exact), and each pair is dependent only on the previous one,
this sequence is eventually periodic. Also, by the Fibonacci relation, one
can find the previous pair to a given pair, so this sequence is immediately
periodic. But F0 ≡ 0 (mod 104 ), so within 108 terms, another Fibonacci
number divisible by 104 must appear.
13
In fact, a computer check shows that 104 | F7500 , and (Fn ) modulo 104
has period 15000, which is much smaller than the upper bound of 108 .
If ax ≡ 1 (mod m), then we say that x is the inverse of a modulo m,
denoted by a−1 , and it is unique modulo m.
Theorem 4.2. The inverse of a modulo m exists and is unique iff a is
relatively prime to m.
Proof. If ax ≡ 1 (mod m), then ax = 1+km for some k ⇒ ax−km = 1.
By Corollary 2.2, a and m are relatively prime. Now, if gcd(a, m) = 1, then
by Corollary 2.2, there exist integers x and y such that ax + my = 1 ⇒ ax =
1 − my ⇒ ax ≡ 1 (mod m). The inverse x is unique modulo m, since if x0 is
also an inverse, then ax ≡ ax0 ≡ 1 ⇒ xax ≡ xax0 ≡ x ≡ x0 .
Corollary 4.3. If p is a prime, then the inverse of a modulo p exists and
is unique iff p does not divide a.
Corollary 4.4. If ak ≡ bk (mod m) and k is relatively prime to m, then
a ≡ b (mod m).
Proof. Multiplying both sides by k −1 , which exists by Theorem 4.2,
yields the result.
We say that a set {a1 , a2 , . . . , am } is a complete residue system modulo
m if for all i, 0 ≤ i ≤ m−1, there exists a unique j such that aj ≡ i (mod m).
Example 4.3. Find all positive integers n such that there exist complete
residue systems {a1 , a2 , . . . , an } and {b1 , b2 , . . . , bn } modulo n for which {a1 +
b1 , a2 + b2 , . . . , an + bn } is also a complete residue system.
Solution. The answer is all odd n. First we prove necessity.
For any complete residue system {a1 , a2 , . . . , an } modulo n, we have that
a1 + a2 + · · · + an ≡ n(n + 1)/2 (mod n). So, if all three sets are complete
residue systems, then a1 +a2 +· · ·+an +b1 +b2 +· · ·+bn ≡ n2 +n ≡ 0 (mod n)
and a1 + b1 + a2 + b2 + · · · + an + bn ≡ n(n + 1)/2 (mod n), so n(n + 1)/2 ≡ 0
(mod n). The quantity n(n + 1)/2 is divisible by n iff (n + 1)/2 is an integer,
which implies that n is odd.
Now assume that n is odd. Let ai = bi = i for all i. Then ai + bi = 2i
for all i, and n is relatively prime to 2, so by Corollary 4.4, {2, 4, . . . , 2n} is
a complete residue system modulo n.
Theorem 4.5. Euler’s Theorem. If a is relatively prime to m, then
a
≡ 1 (mod m).
φ(m)
14
Proof. Let a1 , a2 , . . . , aφ(m) be the positive integers less than m that
are relatively prime to m. Consider the integers aa1 , aa2 , . . . , aaφ(m) . We
claim that they are a permutation of the original φ(m) integers ai , modulo
m. For each i, aai is also relatively prime to m, so aai ≡ ak for some k. Since
aai ≡ aaj ⇔ ai ≡ aj (mod m), each ai gets taken to a different ak under
multiplication by a, so indeed they are permuted. Hence,
a1 a2 · · · aφ(m) ≡ (aa1 )(aa2 ) · · · (aaφ(m) )
≡ aφ(m) a1 a2 · · · aφ(m)
⇒ 1 ≡ aφ(m)
(mod m).
Remark. This gives an explicit formula for the inverse of a modulo m:
a−1 ≡ aφ(m)−2 (mod m). Alternatively, one can use the Euclidean algorithm
to find a−1 ≡ x as in the proof of Theorem 4.2.
Corollary 4.6. Fermat’s Little Theorem (FLT). If p is a prime, and p
does not divide a, then ap−1 ≡ 1 (mod p).
Example 4.4. Show that if a and b are relatively prime positive integers,
then there exist integers m and n such that am + bn ≡ 1 (mod ab).
Solution. Let S = am + bn , where m = φ(b) and n = φ(a). Then
by Euler’s Theorem, S ≡ bφ(a) ≡ 1 (mod a), or S − 1 ≡ 0 (mod a), and
S ≡ aφ(b) ≡ 1 (mod b), or S − 1 ≡ 0 (mod b). Therefore, S − 1 ≡ 0, or S ≡ 1
(mod ab).
Example 4.5. For all positive integers i, let Si be the sum of the products
of 1, 2, . . . , p − 1 taken i at a time, where p is an odd prime. Show that
S1 ≡ S2 ≡ · · · ≡ Sp−2 ≡ 0 (mod p).
Solution. First, observe that
(x − 1)(x − 2) · · · (x − (p − 1))
= xp−1 − S1 xp−2 + S2 xp−3 − · · · − Sp−2 x + Sp−1 .
This polynomial vanishes for x = 1, 2, . . . , p − 1. But by Fermat’s Little
Theorem, so does xp−1 − 1 modulo p. Taking the difference of these two
polynomials, we obtain another polynomial of degree p − 2 with p − 1 roots
modulo p, so it must be the zero polynomial, and the result follows from
comparing coefficients.
15
Remark. We immediately have that (p − 1)! ≡ Sp−1 ≡ −1 (mod p),
which is Wilson’s Theorem. Also, xp − x ≡ 0 (mod p) for all x, yet we
cannot compare coefficients here. Why not?
Theorem 4.7. If p is a prime and n is an integer such that p | (4n2 + 1),
then p ≡ 1 (mod 4).
Proof. Clearly, p cannot be 2, so we need only show that p 6≡ 3 (mod 4).
Suppose p = 4k + 3 for some k. Let y = 2n, so by Fermat’s Little Theorem,
y p−1 ≡ 1 (mod p), since p does not divide n. But, y 2 + 1 ≡ 0, so
y p−1 ≡ y 4k+2 ≡ (y 2 )2k+1 ≡ (−1)2k+1 ≡ −1
(mod p),
contradiction. Therefore, p ≡ 1 (mod 4).
Remark. The same proof can be used to show that if p is a prime and
p | (n2 + 1), then p = 2 or p ≡ 1 (mod 4).
Example 4.6. Show that there are an infinite number of primes of the
form 4k + 1 and of the form 4k + 3.
Solution. Suppose that there are a finite number of primes of the form
4k + 1, say p1 , p2 , . . . , pn . Let N = 4(p1 p2 · · · pn )2 + 1. By Theorem 4.7, N
is only divisible by primes of the form 4k + 1, but clearly N is not divisible
by any of these primes, contradiction.
Similarly, suppose that there are a finite number of primes of the form
4k + 3, say q1 , q2 , . . . , qm . Let M = 4q1 q2 · · · qm − 1. Then M ≡ 3 (mod 4),
so M must be divisible by a prime of the form 4k + 3, but M is not divisible
by any of these primes, contradiction.
Example 4.7. Show that if n is an integer greater than 1, then n does
not divide 2n − 1.
Solution. Let p be the least prime divisor of n. Then gcd(n, p − 1) = 1,
and by Corollary 2.2, there exist integers x and y such that nx+(p−1)y = 1.
If p | (2n − 1), then 2 ≡ 2nx+(p−1)y ≡ (2n )x (2p−1 )y ≡ 1 (mod p) by Fermat’s
Little Theorem, contradiction. Therefore, p - (2n − 1) ⇒ n - (2n − 1).
Theorem 4.8. Wilson’s Theorem. If p is a prime, then (p − 1)! ≡ −1
(mod p). (See also Example 4.5.)
Proof. Consider the congruence x2 ≡ 1 (mod p). Then x2 − 1 ≡ (x −
1)(x + 1) ≡ 0, so the only solutions are x ≡ 1 and −1. Therefore, for each i,
2 ≤ i ≤ p − 2, there exists a unique inverse j 6= i of i, 2 ≤ j ≤ p − 2, modulo
16
p. Hence, when we group in pairs of inverses,
(p − 1)! ≡ 1 · 2 · · · (p − 2) · (p − 1)
≡ 1 · 1 · · · 1 · (p − 1)
≡ −1 (mod p).
Example 4.8. Let {a1 , a2 , . . . , a101 } and {b1 , b2 , . . . , b101 } be complete
residue systems modulo 101. Can {a1 b1 , a2 b2 , . . . , a101 b101 } be a complete
residue system modulo 101?
Solution. The answer is no. Suppose that {a1 b1 , a2 b2 , . . . , a101 b101 } is
a complete residue system modulo 101. Without loss of generality, assume
that a101 ≡ 0 (mod 101). Then b101 ≡ 0 (mod 101), because if any other
bj was congruent to 0 modulo 101, then aj bj ≡ a101 b101 ≡ 0 (mod 101),
contradiction. By Wilson’s Theorem, a1 a2 · · · a100 ≡ b1 b2 · · · b100 ≡ 100! ≡
−1 (mod 101), so a1 b1 a2 b2 · · · a100 b100 ≡ 1 (mod 101). But a101 b101 ≡ 0
(mod 101), so a1 b1 a2 b2 · · · a100 b100 ≡ 100! ≡ −1 (mod 101), contradiction.
Theorem 4.9. If p is a prime, then the congruence x2 + 1 ≡ 0 (mod p)
has a solution iff p = 2 or p ≡ 1 (mod 4). (Compare to Theorem 7.1)
Proof. If p = 2, then x = 1 is a solution. If p ≡ 3 (mod 4), then by the
remark to Theorem 4.7, no solutions exist. Finally, if p = 4k + 1, then let
x = 1 · 2 · · · (2k). Then
x2 ≡ 1 · 2 · · · (2k) · (2k) · · · 2 · 1
≡ 1 · 2 · · · (2k) · (−2k) · · · (−2) · (−1) (multiplying by 2k −1s)
≡ 1 · 2 · · · (2k) · (p − 2k) · · · (p − 2) · (p − 1)
≡ (p − 1)! ≡ −1 (mod p).
Theorem 4.10. Let p be a prime such that p ≡ 1 (mod 4). Then there
exist positive integers x and y such that p = x2 + y 2 .
Proof. By Theorem 4.9, there exists an integer a such that a2 ≡ −1
(mod p). Consider the set of integers of the form ax − y, where x and y
√
are integers, 0 ≤ x, y < p. The number of possible pairs (x, y) is then
√
√
(b pc + 1)2 > ( p)2 = p, so by pigeonhole principle, there exist integers
√
0 ≤ x1 , x2 , y1 , y2 < p, such that ax1 −y1 ≡ ax2 −y2 (mod p). Let x = x1 −x2
and y = y1 − y2 . At least one of x and y is non-zero, and ax ≡ y ⇒ a2 x2 ≡
17
−x2 ≡ y 2 ⇒ x2 + y 2 ≡ 0 (mod p). Thus, x2 + y 2 is a multiple of p, and
√
√
0 < x2 + y 2 < ( p)2 + ( p)2 = 2p, so x2 + y 2 = p.
Theorem 4.11. Let n be a positive integer. Then there exist integers
x and y such that n = x2 + y 2 iff each prime factor of n of the form 4k + 3
appears an even number of times.
Theorem 4.12. The Chinese Remainder Theorem (CRT). If a1 , a2 , . . . ,
ak are integers, and m1 , m2 , . . . , mk are pairwise relatively prime integers,
then the system of congruences
x ≡ a1
x ≡ a2
..
.
x ≡ ak
(mod m1 ),
(mod m2 ),
(mod mk )
has a unique solution modulo m1 m2 · · · mk .
Proof. Let m = m1 m2 · · · mk , and consider m/m1 . This is relatively
prime to m1 , so there exists an integer t1 such that t1 · m/m1 ≡ 1 (mod m1 ).
Accordingly, let s1 = t1 · m/m1 . Then s1 ≡ 1 (mod m1 ) and s1 ≡ 0
(mod mj ), j 6= 1. Similarly, for all i, there exists an si such that si ≡ 1
(mod mi ) and si ≡ 0 (mod mj ), j 6= i. Then, x = a1 s1 + a2 s2 + · · · + ak sk is
a solution to the above system. To see uniqueness, let x0 be another solution.
Then x − x0 ≡ 0 (mod mi ) for all i ⇒ x − x0 ≡ 0 (mod m1 m2 · · · mk ).
Remark. The proof shows explicitly how to find the solution x.
Example 4.9. For a positive integer n, find the number of solutions of
the congruence x2 ≡ 1 (mod n).
Solution. Let the prime factorization of n be 2e pe11 pe22 · · · pekk . By CRT,
x2 ≡ 1 (mod n) ⇔ x2 ≡ 1 (mod pei i ) for all i, and x2 ≡ 1 (mod 2e ). We
consider these cases separately.
We have that x2 ≡ 1 (mod pei i ) ⇔ x2 − 1 = (x − 1)(x + 1) ≡ 0 (mod pei i ).
But pi cannot divide both x − 1 and x + 1, so it divides one of them; that is,
x ≡ ±1 (mod pei i ). Hence, there are two solutions.
Now, if (x − 1)(x + 1) ≡ 0 (mod 2e ), 2 can divide both x − 1 and x + 1,
but 4 cannot divide both. For e = 1 and e = 2, it is easily checked that there
are 1 and 2 solutions respectively. For e ≥ 3, since there is at most one factor
18
of 2 in one of x − 1 and x + 1, there must be at least e − 1 in the other, for
their product to be divisible by 2e . Hence, the only possibilities are x − 1 or
x + 1 ≡ 0, 2e−1 (mod 2e ), which lead to the four solutions x ≡ 1, 2e−1 − 1,
2e−1 + 1, and 2e − 1.
Now that we know how many solutions each prime power factor contributes, the number of solutions modulo n is simply the product of these,
by CRT. The following table gives the answer:
e Number of solutions
0, 1
2k
2
2k+1
≥3
2k+2
Theorem 4.11. Let m be a positive integer, let a and b be integers, and
let k = gcd(a, m). Then the congruence ax ≡ b (mod m) has k solutions or
no solutions according as k | b or k - b.
Problems
1. Prove that for each positive integer n there exist n consecutive positive
integers, none of which is an integral power of a prime.
(1989 IMO)
2. For an odd positive integer n > 1, let S be the set of integers x,
1 ≤ x ≤ n, such that both x and x + 1 are relatively prime to n. Show
that
Y
x ≡ 1 (mod n).
x∈S
3. Find all positive integer solutions to 3x + 4y = 5z .
(1991 IMO Short List)
4. Let n be a positive integer such that n + 1 is divisible by 24. Prove
that the sum of all the divisors of n is divisible by 24.
(1969 Putnam Mathematical Competition)
5. (Wolstenholme’s Theorem) Prove that if
1+
1
1 1
+ + ··· +
2 3
p−1
19
is expressed as a fraction, where p ≥ 5 is a prime, then p2 divides the
numerator.
6. Let a be the greatest positive root of the equation x3 − 3x2 + 1 = 0.
Show that ba1788 c and ba1988 c are both divisible by 17.
(1988 IMO Short List)
7. Let {a1 , a2 , . . . , an } and {b1 , b2 , . . . , bn } be complete residue systems
modulo n, such that {a1 b1 , a2 b2 , . . . , an bn } is also a complete residue
system modulo n. Show that n = 1 or 2.
8. Let m, n be positive integers. Show that 4mn − m − n can never be a
square.
(1984 IMO Proposal)
5
Binomial Coefficients
For non-negative integers n and k, k ≤ n, the binomial coefficient nk is
defined as
n!
,
k!(n − k)!
and has several important properties. By convention, nk = 0 if k > n.
In the following results, for polynomials f and g with integer coefficients,
we say that f ≡ g (mod m) if m divides every coefficient in f − g.
Theorem 5.1. If p is a prime, then the number of factors of p in n! is
n
n
n
+ 2 + 3 + ··· .
p
p
p
It is also
n − sn
,
p−1
where sn is the sum of the digits of n when expressed in base p.
Theorem 5.2. If p is a prime, then
p
≡ 0 (mod p)
i
20
for 1 ≤ i ≤ p − 1.
Corollary 5.3. (1 + x)p ≡ 1 + xp (mod p).
Lemma 5.4. For all real numbers x and y, bx + yc ≥ bxc + byc.
Proof. x ≥ bxc ⇒ x + y ≥ bxc + byc ∈ Z, so bx + yc ≥ bxc + byc.
Theorem 5.5. If p is a prime, then
k
p
≡ 0 (mod p)
i
for 1 ≤ i ≤ pk − 1.
Proof. By Lemma 5.4,
k
X
k k k
X
i
p −i
p
+
≤
,
j
j
p
p
pj
j=1
j=1
where the LHS and RHS
number
of factors
of p in i!(pk − i)! and
j kare jthe
k
j
k
k
k
pk ! respectively. But, pik = p p−i
= 0 and ppk = 1, so the inequality is
k
k
strict, and at least one factor of p divides pi .
k
k
Corollary 5.6. (1 + x)p ≡ 1 + xp (mod p).
Example 5.1. Let n be a positive integer. Show that the product of n
consecutive positive integers is divisible by n!.
Solution. If the consecutive integers are m, m + 1, . . . , m + n − 1, then
m(m + 1) · · · (m + n − 1)
m+n−1
=
.
n!
n
Example 5.2. Let n be a positive integer. Show that
n
n
n
(n + 1) lcm
,
,...,
= lcm(1, 2, . . . , n + 1).
0
1
n
(AMM E2686)
Solution. Let p be a prime ≤ n + 1 and let α (respectively β) be the
highest power of p in the LHS (respectively RHS) of the above equality.
Choose r so that pr ≤ n + 1 < pr+1 . Then clearly β = r. We claim that
m
r
r+1
r+1
if p ≤ m < p , then p
for 0 ≤ k ≤ m.
(∗)
k
21
m
k
Indeed, the number of factors of p in
γ=
r X
m
s=1
ps
is
k
m−k
− s −
.
p
ps
Since each summand in this sum is 0 or 1, we have γ ≤ r; that is, (*) holds.
For 0 ≤ k ≤ n, let
n
n+1
n+1
ak = (n + 1)
= (n − k + 1)
= (k + 1)
.
k
k
k+1
n+1
By (∗), pr+1 does not divide any of the integers nk , n+1
,
or
. Thus,
k
k+1
pr+1 can divide ak only if p divides each of the integers n + 1, n − k + 1, and
k + 1. This implies that p divides (n + 1) − (n − k + 1) − (k + 1) = −1,
r
contradiction. Therefore, pr+1 - ak . On the
other hand, forr k = p − 1,
n+1
we have that k ≤ n and ak = (k + 1) k+1 is divisible by p . Therefore,
β = r = α.
Theorem 5.7. Lucas’s Theorem. Let m and n be non-negative integers,
and p a prime. Let
m = mk pk + mk−1 pk−1 + · · · + m1 p + m0 ,
k
n = nk p + nk−1 p
k−1
and
+ · · · + n1 p + n0
be the base p expansions of m and n respectively. Then
m
mk
mk−1
m1
m0
≡
···
(mod p).
n
nk
nk−1
n1
n0
Proof. By Corollary 5.6,
k +m
k−1 +···+m p+m
1
0
k−1 p
(1 + x)m ≡ (1 + x)mk p
km
k
≡ (1 + x)p
k
k−1 m
k−1
(1 + x)p
k−1
≡ (1 + xp )mk (1 + xp
· · · (1 + x)pm1 (1 + x)m0
)mk−1 · · · (1 + xp )m1 (1 + x)m0
By base p expansion, the coefficient of xn on both sides is
m
mk
mk−1
m1
m0
≡
···
(mod p).
n
nk
nk−1
n1
n0
22
(mod p).
Corollary 5.8. Let n be a positive integer. Let A(n) denote the number
of factors of 2 in n!, and let B(n) denote the number of 1s in the binary
expansion of n. Then the number of odd entries in the nth row of Pascal’s
Triangle, or equivalently the number of odd coefficients in the expansion of
(1 + x)n , is 2B(n) . Furthermore, A(n) + B(n) = n for all n.
Useful Facts
• For a polynomial f with integer coefficients and prime p,
n
n
[f (x)]p ≡ f (xp )
(mod p).
Problems
1. Let a and b be non-negative integers, and p a prime. Show that
pa
a
≡
(mod p).
pb
b
2. Let an be the last non-zero digit in the decimal representation of the
number n!. Is the sequence a1 , a2 , a3 , . . . eventually periodic?
(1991 IMO Short List)
3. Find all positive integers n such that 2n | (3n − 1).
4. Find the greatest integer k for which 1991k divides
19901991
1992
+ 19921991
1990
.
(1991 IMO Short List)
5. For a positive integer n, let a(n) and b(n) denote the number of binomial
coefficients in the nth row of Pascal’s Triangle that are congruent to 1
and 2 modulo 3 respectively. Prove that a(n) − b(n) is always a power
of 2.
6. Let n be a positive integer. Prove that if the number of factors of 2 in
n! is n − 1, then n is a power of 2.
23
7. For a positive integer n, let
1
2n
Cn =
,
n+1 n
and Sn = C1 + C2 + · · · + Cn .
Prove that Sn ≡ 1 (mod 3) if and only if there exists a 2 in the base 3
expansion of n + 1.
6
Order of an Element
We know that if a is relatively prime to m, then there exists a positive integer
n such that an ≡ 1 (mod m). Let d be the smallest such n. Then we say
that d is the order of a modulo m, denoted by ordm (a), or simply ord(a) if
the modulus m is understood.
Theorem 6.1. If a is relatively prime to m, then an ≡ 1 (mod m) iff
ord(a) | n. Furthermore, an0 ≡ an1 iff ord(a) | (n0 − n1 ).
Proof. Let d = ord(a). It is clear that d | n ⇒ an ≡ 1 (mod m). On
the other hand, if an ≡ 1 (mod m), then by the division algorithm, there
exist integers q and r such that n = qd + r, 0 ≤ r < d. Then an ≡ aqd+r ≡
(ad )q ar ≡ ar ≡ 1 (mod m). But r < d, so r = 0 ⇒ d | n. The second part of
the theorem follows.
Remark. In particular, by Euler’s Theorem, ord(a) | φ(m).
Example 6.1. Show that the order of 2 modulo 101 is 100.
Solution. Let d = ord(2). Then d | φ(101), or d | 100. If d < 100,
then d divides 100/2 or 100/5; that is, d is missing at least one prime factor.
However,
250 ≡ 10245 ≡ 145 ≡ 196 · 196 · 14 ≡ (−6) · (−6) · 14 ≡ −1
(mod 101),
and
220 ≡ 10242 ≡ 142 ≡ −6
(mod 101),
so d = 100.
Example 6.2. Prove that if p is a prime, then every prime divisor of
2p − 1 is greater than p.
24
Solution. Let q | (2p − 1), where q is a prime. Then 2p ≡ 1 (mod q), so
ord(2) | p. But ord(2) 6= 1, so ord(2) = p. And by Fermat’s Little Theorem,
ord(2) | (q − 1) ⇒ p ≤ q − 1 ⇒ q > p.
In fact, for p > 2, q must be of the form 2kp + 1. From the above,
ord(2) | (q − 1), or p | (q − 1) ⇒ q = mp + 1. Since q must be odd, m must
be even.
Example 6.3. Let p be a prime that is relatively prime to 10, and let n
be an integer, 0 < n < p. Let d be the order of 10 modulo p.
(a) Show that the length of the period of the decimal expansion of n/p is
d.
(b) Prove that if d is even, then the period of the decimal expansion of n/p
can be divided into two halves, whose sum is 10d/2 − 1. For example,
1/7 = 0.142857, so d = 6, and 142 + 857 = 999 = 103 − 1.
Solution. (a) Let m be the length of the period, and let n/p =
0.a1 a2 . . . am . Then
10m n
= a1 a2 . . . am .a1 a2 . . . am
p
(10m − 1) n
⇒
= a1 a2 . . . am ,
p
an integer. Since n and p are relatively prime, p must divide 10m − 1, so d
divides m. Conversely, p divides 10d −1, so (10d −1)n/p is an integer, with at
most d digits. If we divide this integer by 10d − 1, then we obtain a rational
number, whose decimal expansion has period at most d. Therefore, m = d.
(b) Let d = 2k, so n/p = 0.a1 a2 . . . ak ak+1 . . . a2k . Now p divides 10d −1 =
102k − 1 = (10k − 1)(10k + 1). However, p cannot divide 10k − 1 (since the
order of 10 is 2k), so p divides 10k + 1. Hence,
10k n
= a1 a2 . . . ak .ak+1 . . . a2k
p
(10k + 1) n
⇒
= a1 a2 . . . ak + 0.a1 a2 . . . ak + 0.ak+1 . . . a2k
p
is an integer. This can occur iff a1 a2 . . . ak +ak+1 . . . a2k is a number consisting
only of 9s, and hence, equal to 10k − 1.
Problems
25
1. Prove that for all positive integers a > 1 and n, n | φ(an − 1).
2. Prove that if p is a prime, then pp −1 has a prime factor that is congruent
to 1 modulo p.
3. For any integer a, set na = 101a−100·2a . Show that for 0 ≤ a, b, c, d ≤
99, na + nb ≡ nc + nd (mod 10100) implies {a, b} = {c, d}.
(1994 Putnam Mathematical Competition)
n
4. Show that if 3 ≤ d ≤ 2n+1 , then d - (a2 + 1) for all positive integers a.
7
Quadratic Residues
Let m be an integer greater than 1, and a an integer relatively prime to m. If
x2 ≡ a (mod m) has a solution, then we say that a is a quadratic residue
of m. Otherwise, we say that a is a quadratic non-residue. Now, let p be
an odd prime. Then the Legendre symbol
a
p
is assigned the value of 1 if a is a quadratic residue of p. Otherwise, it is
assigned the value of −1.
Theorem 7.1. Let p be an odd prime, and a and b be integers relatively
prime to p. Then
(a) ap ≡ a(p−1)/2 (mod p), and
(b)
a
p
b
p
=
ab
p
.
Proof. If the congruence x2 ≡ a (mod p) has a solution, then a(p−1)/2 ≡
xp−1 ≡ 1 (mod p), by Fermat’s Little Theorem. If the congruence x2 ≡ a
(mod p) has no solutions, then for each i, 1 ≤ i ≤ p − 1, there is a unique
j 6= i, 1 ≤ j ≤ p − 1, such that ij ≡ a. Therefore, all the integers from 1 to
p − 1 can be arranged into (p − 1)/2 such pairs. Taking their product,
a(p−1)/2 ≡ 1 · 2 · · · (p − 1) ≡ (p − 1)! ≡ −1
26
(mod p),
by Wilson’s Theorem. Part (b) now follows from part (a).
Remark. Part (a) is known as Euler’s criterion.
Example 7.1. Show that if p is an odd prime, then
1
2
p−1
+
+ ··· +
= 0.
p
p
p
Solution. Note that 12 , 22 , . . . , ((p − 1)/2)2 are distinct modulo p,
and that ((p + 1)/2)2 , . . . , (p − 1)2 represent the same residues, simply in
reverse. Hence, there are exactly (p−1)/2 quadratic residues, leaving (p−1)/2
quadratic non-residues. Therefore, the given sum contains (p − 1)/2 1s and
(p − 1)/2 −1s.
Theorem 7.2. Gauss’s Lemma. Let p be an odd prime and let a be
relatively prime to p. Consider the least non-negative residues of a, 2a, . . . ,
((p − 1)/2)a modulo p. If n is the number of these residues that are greater
than p/2, then
a
= (−1)n .
p
Theorem 7.3. If p is an odd prime, then −1
= (−1)(p−1)/2 ; that is,
p
−1
p
=
1 if p ≡ 1
−1 if p ≡ 3
(mod 4),
(mod 4).
Proof. This follows from Theorem 4.9 (and Theorem 7.1).
2
Theorem 7.4. If p is an odd prime, then p2 = (−1)(p −1)/8 ; that is,
2
1 if p ≡ 1 or 7 (mod 8),
=
−1 if p ≡ 3 or 5 (mod 8).
p
27
Proof. If p ≡ 1 or 5 (mod 8), then
p−1
(p−1)/2
2
! ≡ 2 · 4 · 6 · · · (p − 1)
2
p−1
p−3
≡ 2 · 4 · 6···
· −
· · · (−5) · (−3) · (−1)
2
2
p−1
(p−1)/4
!
≡ (−1)
2
⇒ 2(p−1)/2 ≡ (−1)(p−1)/4 (mod p).
By Theorem 7.1, p2 = (−1)(p−1)/4 . Hence, p2 = 1 or −1 according as
p ≡ 1 or 5 (mod 8).
Similarly, if p ≡ 3 or 7 (mod 8), then
p−3
p−1
p−1
(p−1)/2
2
! ≡ 2 · 4 · 6···
· −
· · · (−5) · (−3) · (−1)
2
2
2
p−1
(p+1)/4
≡ (−1)
!
2
⇒ 2(p−1)/2 ≡ (−1)(p+1)/4 (mod p).
Hence, p2 = 1 or −1 according as p ≡ 7 or 3 (mod 8).
Example 7.2. Prove that if n is an odd positive integer, then every
prime divisor of 2n − 1 is of the form 8k ± 1. (Compare to Example 6.2)
Solution. Let p | (2n − 1), where pis prime. Let n = 2m + 1. Then
2n ≡ 22m+1 ≡ 2(2m )2 ≡ 1 (mod p) ⇒ p2 = 1 ⇒ p is of the form 8k ± 1.
Theorem 7.5. The Law of Quadratic Reciprocity. For distinct odd
primes p and q,
p−1 q−1
q
p
= (−1) 2 · 2 .
q
p
Example 7.3. For which primes p > 3 does the congruence x2 ≡ −3
(mod p) have a solution?
−1
3
Solution. We seek p for which −3
=
= 1. By quadratic
p
p
p
reciprocity,
3
p
−1
(p−1)/2
= (−1)
=
,
p
3
p
28
by Theorem 7.3. Thus, in general,
2 −1
3
p
−1
−3
p
=
=
.
=
p
p
p
3
p
3
And, ( p3 ) = 1 iff p ≡ 1 (mod 3). Since p 6≡ 4 (mod 6), we have that x2 ≡ −3
(mod p) has a solution iff p ≡ 1 (mod 6).
Example 7.4. Show that if p = 2n + 1, n ≥ 2, is prime, then 3(p−1)/2 + 1
is divisible by p.
Solution. We must have that n is even, say 2k, for otherwise p ≡ 0
(mod 3). By Theorem 7.1,
3
≡ 3(p−1)/2 (mod p).
p
However, p ≡ 1 (mod 4), and p ≡ 4k + 1 ≡ 2 (mod 3) ⇒ p3 = −1, and by
quadratic reciprocity,
p
3
= (−1)(p−1)/2 = 1,
p
3
so
3
= −1 ⇒ 3(p−1)/2 + 1 ≡ 0
p
(mod p).
Useful Facts
• (a) If p is a prime and p ≡ 1 or 3 (mod 8), then there exist positive
integers x and y such that p = x2 + 2y 2 .
(b) If p is a prime and p ≡ 1 (mod 6), then there exist positive integers
x and y such that p = x2 + 3y 2 .
Problems
1. Show that if p > 3 is a prime, then the sum of the quadratic residues
among the integers 1, 2, . . . , p − 1 is divisible by p.
2. Let Fn denote the nth Fibonacci number. Prove that if p > 5 is a prime,
then
p
Fp ≡
(mod p).
5
29
3. Show that 16 is a perfect 8th power modulo p for any prime p.
4. Let a, b, and c be positive integers that are pairwise relatively prime,
and that satisfy a2 − ab + b2 = c2 . Show that every prime factor of c is
of the form 6k + 1.
5. Let p be an odd prime and let ζ be a primitive pth root of unity; that is,
ζ is a complex number such that ζ p = 1 and ζ k 6= 1 for 1 ≤ k ≤ p − 1.
Let Ap and Bp denote the set of quadratic
and non-residues
P residues
P
k
modulo p, respectively. Finally, let α = k∈Ap ζ and β = k∈Bp ζ k .
For example, for p = 7, α = ζ + ζ 2 + ζ 4 and β = ζ 3 + ζ 5 + ζ 6 . Show
that α and β are the roots of
p
1 − −1
p
= 0.
x2 + x +
4
8
Primitive Roots
If the order of g modulo m is φ(m), then we say that g is a primitive root
modulo m, or simply of m.
Example 8.1. Show that 2 is a primitive root modulo 3n for all n ≥ 1.
Solution. The statement is easily verified for n = 1, so assume the result
k
k−1
is true for some n = k; that is, 2φ(3 ) ≡ 22·3
≡ 1 (mod 3k ). Now, let d be
the order of 2 modulo 3k+1 . Then 2d ≡ 1 (mod 3k+1 ) ⇒ 2d ≡ 1 (mod 3k ),
so 2 · 3k−1 | d. However, d | φ(3k+1 ), or d | 2 · 3k . We deduce that d is either
2 · 3k−1 or 2 · 3k . Now we require the following lemma:
n−1
≡ 1 + 3n (mod 3n+1 ), for all n ≥ 1.
Lemma. 22·3
This is true for n = 1, so assume it is true for some n = k. Then by
assumption,
22·3
k−1
= 1 + 3k + 3k+1 m for some integer m
k
⇒ 22·3 = 1 + 3k+1 + 3k+2 M
2·3k
⇒ 2
≡ 1 + 3k+1
for some integer M (obtained by cubing)
(mod 3k+2 ).
By induction, the lemma is proved.
k−1
Therefore, 22·3 ≡ 1 + 3k 6≡ 1 (mod 3k+1 ), so the order of 2 modulo 3k+1
is 2 · 3k , and again by induction, the result follows.
30
Corollary 8.2. If 2n ≡ −1 (mod 3k ), then 3k−1 | n.
Proof. The given implies 22n ≡ 1 (mod 3k ) ⇒ φ(3k ) | 2n, or 3k−1 | n.
Theorem 8.3. If m has a primitive root, then it has φ(φ(m)) (distinct)
primitive roots modulo m.
Theorem 8.4. The positive integer m has a primitive root iff m is one
of 2, 4, pk , or 2pk , where p is an odd prime.
Theorem 8.5. If g is a primitive root of m, then g n ≡ 1 (mod m) iff
φ(m) | n. Furthermore, g n0 ≡ g n1 iff φ(m) | (n0 − n1 ).
Proof. This follows directly from Theorem 6.1.
Theorem 8.6. If g is a primitive root of m, then the powers 1, g, g 2 ,
. . . , g φ(m)−1 represent each integer relatively prime to m uniquely modulo m.
In particular, if m > 2, then g φ(m)/2 ≡ −1 modulo m.
Proof. Clearly, each power g i is relatively prime to m, and there are
φ(m) integers relatively prime to m. Also, if g i ≡ g j (mod m), then g i−j ≡
1 ⇒ φ(m) | (i − j) by Theorem 8.6, so each of the powers are distinct
modulo m. Hence, each integer relatively prime to m is some power g i
modulo m. Furthermore, there is a unique i, 0 ≤ i ≤ φ(m) − 1, such that
g i ≡ −1 ⇒ g 2i ≡ 1 ⇒ 2i = φ(m), or i = φ(m)/2.
Proposition 8.7. Let m be a positive integer. Then the only solutions to
the congruence x2 ≡ 1 (mod m) are x ≡ ±1 (mod m) iff m has a primitive
root.
Proof. This follows from Example 4.9.
Example 8.2. For a positive integer m, let S be the set of positive
integers less than m that are relatively prime to m, and let P be the product
of the elements in S. Show that P ≡ ±1 (mod m), with P ≡ −1 (mod m)
iff m has a primitive root.
Solution. We use a similar strategy as in the proof of Wilson’s Theorem.
The result is clear for m = 2, so assume that m ≥ 3. We partition S as
follows: Let A be the elements of S that are solutions to the congruence
x2 ≡ 1 (mod m), and let B be the remaining elements. The elements in B
can be arranged into pairs, by pairing each with its distinct multiplicative
inverse. Hence, the product of the elements in B is 1 modulo m.
The elements in A may also be arranged into pairs, by pairing each with
31
its distinct additive inverse, i.e. x and m − x. These must be distinct,
because otherwise, x = m/2, which is not relatively prime to m. Note that
their product is x(m − x) ≡ mx − x2 ≡ −1 (mod m). Now if m has a
primitive root, then by Proposition 8.7, A consists of only the two elements
1 and −1, so P ≡ −1 (mod m). Otherwise, by Example 4.9, the number of
elements of A is a power of two that is at least 4, so the number of such pairs
in A is even, and P ≡ 1 (mod m).
Remark. For m prime, this simply becomes Wilson’s Theorem.
Theorem 8.8.
(1) If g is a primitive root of p, p a prime, then g or g + p is a primitive
root of p2 , according as g p−1 6≡ 1 (mod p2 ) or g p−1 ≡ 1 (mod p2 ).
(2) If g is a primitive root of pk , where k ≥ 2 and p is prime, then g is a
primitive root of pk+1 .
By Theorem 8.6, given a primitive root g of m, for each a relatively prime
to m, there exists a unique integer i modulo φ(m) such that g i ≡ a (mod m).
This i is called the index of a with respect to the base g, denoted by indg (a)
(i is dependent on g, so it must be specified). Indices have striking similarity
to logarithms, as seen in the following properties:
(1) indg (1) ≡ 0 (mod φ(m)), indg (g) ≡ 1 (mod φ(m)),
(2) a ≡ b (mod m) ⇒ indg (a) ≡ indg (b) (mod φ(m)),
(3) indg (ab) ≡ indg (a) + indg (b) (mod φ(m)),
(4) indg (ak ) ≡ k indg (a) (mod φ(m)).
Theorem 8.9. If p is a prime and a is not divisible by p, then the congruence xn ≡ a (mod p) has gcd(n, p − 1) solutions or no solutions according
as
a(p−1)/ gcd(n,p−1) ≡ 1
(mod p) or a(p−1)/ gcd(n,p−1) 6≡ 1
(mod p).
Proof. Let g be a primitive root of p, and let i be the index of a with
respect to g. Also, any solution x must be relatively prime to p, so let u be
the index of x. Then the congruence xn ≡ a becomes g nu ≡ g i (mod p) ⇔
nu ≡ i (mod p − 1). Let k = gcd(n, p − 1). Since g is a primitive root of p,
k | i ⇔ g i(p−1)/k ≡ a(p−1)/k ≡ 1. The result now follows from Theorem 4.11.
32
Remark. Taking p to be an odd prime and n = 2, we deduce Euler’s
criterion.
Example 8.3 Let n ≥ 2 be an integer and p = 2n + 1. Show that if
3(p−1)/2 + 1 ≡ 0 (mod p), then p is a prime. (The converse to Example 7.4.)
n
n−1
≡ −1 (mod p), we obtain 32 ≡ 1
Solution. From 3(p−1)/2 ≡ 32
n
(mod p), so the order of 3 is 2 = p−1, but the order also divides φ(p) ≥ p−1.
Therefore, φ(p) = p − 1, and p is a prime.
Example 8.4. Prove that if n = 3k−1 , then 2n ≡ −1 (mod 3k ). (A
partial converse to Corollary 8.2.)
Solution. By Example 8.1, 2 is a primitive root of 3k . Therefore, 2 has
order φ(3k ) = 2 · 3k−1 = 2n ⇒ 22n ≡ 1 ⇒ (2n − 1)(2n + 1) ≡ 0 (mod 3k ).
k−1
However, 2n − 1 ≡ (−1)3 − 1 ≡ 1 6≡ 0 (mod 3), so 2n + 1 ≡ 0 (mod 3k ).
Example 8.5. Find all positive integers n > 1 such that
2n + 1
n2
is an integer.
(1990 IMO)
Solution. Clearly, n must be odd. Now assume that 3k kn; that is, 3k
is the highest power of 3 dividing n. Then 32k | n2 | (2n + 1) ⇒ 2n ≡ −1
(mod 32k ) ⇒ 32k−1 | n, by Corollary 8.2 ⇒ 2k − 1 ≤ k ⇒ k ≤ 1, showing
that n has at most one factor of 3. We observe that n = 3 is a solution.
Suppose that n has a prime factor greater than 3; let p be the least such
prime. Then p | (2n +1) ⇒ 2n ≡ −1 (mod p). Let d be the order of 2 modulo
p. Since 22n ≡ 1, d | 2n. If d is odd, then d | n ⇒ 2n ≡ 1, contradiction,
so d is even, say d = 2d1 . Then 2d1 | 2n ⇒ d1 | n. Also, d | (p − 1), or
2d1 | (p − 1) ⇒ d1 ≤ (p − 1)/2 < p. But d1 | n, so d1 = 1 or d1 = 3. If d1 = 1,
then d = 2, and 22 ≡ 1 (mod p), contradiction. If d1 = 3, then d = 6, and
26 ≡ 1 (mod p), or p | 63 ⇒ p = 7. However, the order of 2 modulo 7 is 3,
which is odd, again contradiction. Therefore, no such p can exist, and the
only solution is n = 3.
Useful Facts
n
• All prime divisors of the Fermat number 22 + 1, n > 1, are of the form
2n+2 k + 1.
33
Problems
1. Let p be an odd prime. Prove that
1i + 2i + · · · + (p − 1)i ≡ 0
(mod p)
for all i, 0 ≤ i ≤ p − 2.
2. Show that if p is an odd prime, then the congruence x4 ≡ −1 (mod p)
has a solution iff p ≡ 1 (mod 8).
n
3. Show that if a and n are positive integers with a odd, then a2 ≡ 1
(mod 2n+2 ).
4. The number 142857 has the remarkable property that multiplying it
by 1, 2, 3, 4, 5, and 6 cyclically permutes the digits. What are other
numbers that have this property? Hint: Compute 142857 × 7.
9
Dirichlet Series
Despite the intimidating name, Dirichlet series are easy to work with, and can
provide quick proofs to certain number-theoretic identities, such as Example
3.2. Let α be a function taking the positive integers to the integers. Then
we say that
f (s) =
∞
X
α(n)
n=1
ns
= α(1) +
α(2) α(3)
+ s + ···
2s
3
is the Dirichlet series generating function (Dsgf ) of the function α,
which we denote by f (s) ↔ α(n). Like general generating functions, these
generating functions are used to provide information about their corresponding number-theoretic functions, primarily through manipulation of the generating functions.
Let 1 denote the function which is 1 for all positive integers; that is,
1(n) = 1 for all n. Let δ1 (n) be the function defined by
1 if n = 1,
δ1 (n) =
0 if n > 1.
It is easy to check that 1 and δ1 are multiplicative.
34
Now, let α and β be functions taking the positive integers to the integers.
The convolution of α and β, denoted α ∗ β, is defined by
X
(α ∗ β)(n) =
α(d)β(n/d).
d|n
Note that convolution is symmetric; that is, α ∗ β = β ∗ α.
Theorem 9.1. Let f (s) ↔ α(n) and g(s) ↔ β(n). Then (f · g)(s) ↔
(α ∗ β)(n).
We now do three examples. The Dsgf of 1(n) is the well-known Riemann
Zeta function ζ(s):
∞
X
1
1
1
= 1 + s + s + ··· ,
ζ(s) =
s
n
2
3
n=1
so ζ(s) ↔ 1(n). This function will play a prominent role in this theory.
What makes this theory nice to work with is that we may work with these
functions at a purely formal level; no knowledge of the analytic properties of
ζ(s) or indeed of any other Dsgf is required.
By Theorem 9.1, the number-theoretic function corresponding to ζ 2 (s) is
X
X
1(d)1(n/d) =
1 = τ (n).
d|n
d|n
Hence, ζ 2 (s) ↔ τ (n). Finally, it is clear that 1 ↔ δ1 (n).
If α is a multiplicative function, then we can compute the Dsgf corresponding to α using the following theorem.
Theorem 9.2. Let α be a multiplicative function. Then
∞
X
α(n)
n=1
ns
=
∞
YX
α(pk )
p
k=0
pks
=
Y
[1 + α(p)p−s + α(p2 )p−2s + α(p3 )p−3s + · · · ],
p
where the product on the right is taken over all prime numbers.
35
As before, if we take α = 1, then we obtain
Y
ζ(s) =
(1 + p−s + p−2s + p−3s + · · · )
p
=
Y
p
=Q
1
1 − p−s
1
,
−s
p (1 − p )
an identity that will be useful.
We say that a positive integer n > 1 is square-free if n contains no
repeated prime factors; that is, p2 - n for all primes p. With this in mind, we
define the Möbius function µ as follows:

if n = 1,
 1
0
if n is not square-free, and
µ(n) =

k
(−1) if n is square-free and has k prime factors.
It is easy to check that µ is multiplicative. By Theorem 9.2, the corresponding
Dsgf is given by
Y
1
.
(1 − p−s ) =
ζ(s)
p
Hence, 1/ζ(s) ↔ µ(n), and this property makes the the seemingly mysterious
function µ very important, as seen in the following theorem.
Theorem 9.3. (Möbius Inversion Formula) Let α and β be functions
such that
X
β(n) =
α(d).
d|n
Then
α(n) =
X
µ(n/d)β(d).
d|n
Proof. Let f (s) ↔ α(n) and g(s) ↔ β(n). The condition is equivalent to
β = α ∗ 1, or g(s) = f (s)ζ(s), and the conclusion is equivalent to α = β ∗ µ,
or f (s) = g(s)/ζ(s).
Theorem 9.4. Let f (s) ↔ α(n). Then for any integer k, f (s − k) ↔
n α(n).
k
36
For more on Dirichlet series, and generating functions in general, see H.
Wilf, Generatingfunctionology.
Problems
1. Let α, β, and γ be functions taking the positive integers to the integers.
(a) Prove that α ∗ δ1 = α.
(b) Prove that (α ∗ β) ∗ γ = α ∗ (β ∗ γ).
(c) Prove that if α and β are multiplicative, then so is α ∗ β.
2. Prove that the following relations hold:
ζ(s − 1)
↔ φ(n),
ζ(s)
ζ(s)ζ(s − 1) ↔ σ(n),
ζ(s)
↔ |µ(n)|.
ζ(2s)
3. Let the prime factorization of a positive integer n > 1 be pe11 pe22 · · · pekk .
Define the functions λ and θ by λ(n) = (−1)e1 +e2 +···+ek and θ(n) = 2k .
Set λ(1) = θ(1) = 1. Show that λ and θ are multiplicative, and that
ζ(2s)
↔ λ(n) and
ζ(s)
ζ 2 (s)
↔ θ(n).
ζ(2s)
4. For all positive integers n, let
f (n) =
(a) Show that f (n) =
(b) Let n =
that
pe11 pe22
· · · pekk
f (n) =
P
d|n
n
X
n
.
gcd(m,
n)
m=1
dφ(d).
> 1 be the prime factorization of n. Show
1 +1
p2e
+1
1
p1 + 1
p22e2 +1 + 1
p2 + 1
5. Verify Example 3.2 in one calculation.
37
···
p12ek +1 + 1
pk + 1
.
6. Let id denote the identity function; that is, id(n) = n for all n. Verify
each of the following identities in one calculation:
(a) φ ∗ τ = σ.
(b) µ ∗ 1 = δ1 .
(c) µ ∗ id = φ.
(d) φ ∗ σ = id · τ .
(e) σ ∗ id = 1 ∗ (id · τ ).
7. Let a1 , a2 , . . . , be the sequence of positive integers satisfying
X
ad = 2n
d|n
for all n. Hence, a1 = 2, a2 = 22 − 2 = 2, a3 = 23 − 2 = 6, a4 =
24 − 2 − 2 = 12, and so on. Show that for all n, n | an .
Hint: Don’t use the Dsgf of (an )∞
1 ; use the Möbius Inversion Formula.
Bigger Hint: Consider the function f : [0, 1] → [0, 1] defined by f (x) =
{2x}, where {x} = x − bxc is the fractional part of x. Find how the
formula in the problem relates to the function f (n) = f ◦ f ◦ · · · ◦ f .
{z
}
|
n
8. For all non-negative integers k, let σk be the function defined by
X
σk (n) =
dk .
d|n
Thus, σ0 = τ and σ1 = σ. Prove that
ζ(s)ζ(s − k) ↔ σk (n).
10
10.1
Miscellaneous Topics
Pell’s Equations
Pell’s equations (or Fermat’s equations, as they are rightly called) are
diophantine equations of the form x2 − dy 2 = N , where d is a positive nonsquare integer. There always exist an infinite number of solutions when
N = 1, which we characterize.
38
Theorem 10.1.1. If (a, b) is the lowest positive integer solution of x2 −
dy = 1, then all positive integer solutions are of the form
√ n
√ n
√ n!
√ n
(a + b d) + (a − b d) (a + b d) − (a − b d)
√
,
.
(xn , yn ) =
2
2 d
2
We will not give a proof here, but we will verify that every pair indicated
by the formula is a solution.
The pair (xn , yn ) satisfy the equations
√
√
xn + yn d = (a + b d)n , and
√
√
xn − yn d = (a − b d)n .
Therefore,
√
√
x2n − dyn2 = (xn + yn d)(xn − yn d)
√
√
= (a + b d)n (a − b d)n
= (a2 − db2 )n
= 1,
since (a, b) is a solution.
Remark. The sequences (xn ), (yn ) satisfy the recurrence relations xn =
2axn−1 − xn−2 , yn = 2ayn−1 − yn−2 .
For x2 − dy 2 = −1, the situation is similar. If (a, b) is the least positive
solution, then the (xn , yn ) as above for n odd are the solutions of x2 − dy 2 =
−1, and the (xn , yn ) for n even are the solutions of x2 − dy 2 = 1.
Example 10.1.1 Find all solutions in pairs of positive integers (x, y) to
the equation x2 − 2y 2 = 1.
Solution. We find that the lowest positive integer solution is (3,2), so
all positive integer solutions are given by
√
√ !
√
√
(3 + 2 2)n + (3 − 2 2)n (3 + 2 2)n − (3 − 2 2)n
√
,
.
(xn , yn ) =
2
2 2
The first few solutions are (3,2), (17,12), and (99,70).
39
Example 10.1.2. Prove that the equation x2 − dy 2 = −1 has no solution
in integers if d ≡ 3 (mod 4).
Solution. It is apparent that d must have a prime factor of the form 4k +
3, say q. Then x2 ≡ −1 (mod q), which by Theorem 4.9 is a contradiction.
Problems
1. In the sequence
1 5 11 27
, ,
,
, ...,
2 3 8 19
the denominator of the nth term (n > 1) is the sum of the numerator
and the denominator of the (n − 1)th term. The numerator of the nth
term is the sum of the denominators of the nth and (n−1)th term. Find
the limit of this sequence.
(1979 Atlantic Region Mathematics League)
2. Let x0 = 0, x1 = 1, xn+1 = 4xn − xn−1 , and y0 = 1, y1 = 2, yn+1 =
4yn − yn−1 . Show for all n ≥ 0 that yn2 = 3x2n + 1.
(1988 Canadian Mathematical Olympiad)
3. The polynomials P , Q are such that deg P = n, deg Q = m, have the
same leading coefficient, and P 2 (x) = (x2 − 1)Q2 (x) + 1. Show that
P 0 (x) = nQ(x).
(1978 Swedish Mathematical Olympiad, Final Round)
10.2
Farey Sequences
The nth Farey sequence is the sequence of all reduced rationals in [0,1],
with both numerator and denominator no greater than n, in increasing order.
Thus, the first 5 Farey sequences are:
0/1,
0/1,
0/1,
0/1,
0/1,
1/5,
1/3,
1/4, 1/3,
1/4, 1/3, 2/5,
1/2,
1/2,
1/2,
1/2, 3/5,
2/3,
2/3, 3/4,
2/3, 3/4,
Properties of Farey sequences include the following:
40
4/5,
1/1,
1/1,
1/1,
1/1,
1/1.
(1) If a/b and c/d are consecutive fractions in the same sequence, in that
order, then ad − bc = 1.
(2) If a/b, c/d, and e/f are consecutive fractions in the same sequence, in
that order, then
c
a+e
= .
b+f
d
(3) If a/b and c/d are consecutive fractions in the same sequence, then
among all fractions between the two, (a + c)/(b + d) (reduced) is the
unique fraction with the smallest denominator.
For proofs of these and other interesting properties, see Ross Honsberger,
“Farey Sequences”, Ingenuity in Mathematics.
Problems
1. Let a1 , a2 , . . . , am be the denominators of the fractions in the nth Farey
sequence, in that order. Prove that
1
1
1
+
+ ··· +
= 1.
a1 a2 a2 a3
am−1 am
10.3
Continued Fractions
Let a0 , a1 , . . . , an be real numbers, all positive, except possibly a0 . Then let
ha0 , a1 , . . . , an i denote the continued fraction
1
a0 +
.
1
a1 + · · · +
an−1 +
1
an
If each ai is an integer, then we say that the continued fraction is simple.
Define sequences (pk ) and (qk ) as follows:
p−1 = 0, p0 = a0 , and pk = ak pk−1 + pk−2 ,
q−1 = 0, q0 = 1, and qk = ak pk−1 + qk−2 ,
Theorem 10.3.1. For all x > 0 and k ≥ 1,
ha0 , a1 , . . . , ak−1 , xi =
41
xpk−1 + pk−2
.
xqk−1 + qk−2
for k ≥ 1.
In particular,
pk
.
qk
ha0 , a1 , . . . , ak i =
Theorem 10.3.2. For all k ≥ 0,
(1) pk qk−1 − pk−1 qk = (−1)k−1 ,
(2) pk qk−2 − pk−2 qk = (−1)k ak .
Define ck to be the k th convergence ha0 , a1 , . . . , ak i = pk /qk .
Theorem 10.3.3. c0 < c2 < c4 < · · · < c5 < c3 < c1 .
For a nice connection between continued fractions, linear diophantine
equations, and Pell’s equations, see Andy Liu, “Continued Fractions and
Diophantine Equations”, Volume 3, Issue 2, Mathematical Mayhem.
Problems
1. Let a = h1, 2, . . . , 99i and b = h1, 2, . . . , 99, 100i. Prove that
|a − b| <
1
.
99!100!
(1990 Tournament of Towns)
2. Evaluate
v
u
u
u
2207 −
8
t
1
.
1
2207 −
2207 − · · ·
Express your answer in the form
√
a+b c
,
d
where a, b, c, d are integers.
(1995 Putnam)
10.4
The Postage Stamp Problem
Let a and b be relatively prime positive integers greater than 1. Consider the
set of integers of the form ax + by, where x and y are non-negative integers.
The following are true:
(1) The greatest integer that cannot be written in the given form is (a −
1)(b − 1) − 1 = ab − a − b.
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(2) There are 12 (a − 1)(b − 1) positive integers that cannot be written in
the given form.
(3) For all integers t, 0 ≤ t ≤ ab − a − b, t can be written in the given form
iff ab − a − b − t cannot be.
(If you have not seen or attempted this enticing problem, it is strongly
suggested you have a try before reading the full solution.)
Before presenting the solution, it will be instructive to look at an example.
Take a = 12 and b = 5. The first few non-negative integers, in rows of 12,
with integers that cannot be written in the given form in bold, are shown:
0
1
2
12 13 14
24 25 26
36 37 38
48 49 50
3 4
15 16
27 28
39 40
51 52
5
6
17 18
29 30
41 42
53 54
7
19
31
43
55
8 9
20 21
32 33
44 45
56 57
10 11
22 23
34 35
46 47
58 59
With this arrangement, one observation should become immediately apparent, namely that bold numbers in each column end when they reach a
multiple of 5. It should be clear that when reading down a column, once
one hits an integer that can be written in the given form, then all successive
integers can be as well, since we are adding 12 for each row we go down. It
will turn out that this one observation is the key to the solution.
Proof. Define a grapefruit to be an integer that may be written in the
given form. For each i, 0 ≤ i ≤ a − 1, let mi be the least non-negative integer
such that b | (i + ami ). It is obvious that for k ≥ mi , i + ak is a grapefruit.
We claim that for 0 ≤ k ≤ mi − 1, i + ak is not a grapefruit. It is sufficient
to show that i + a(mi − 1) is not a grapefruit, if mi ≥ 1.
Let i + ami = bni , ni ≥ 0. Since i + a(mi − b) = b(ni − a), mi must be
strictly less than b; otherwise, we can find a smaller mi . Then i + a(mi − b) ≤
a−1−a = −1, so ni < a, or ni ≤ a−1. Suppose that ax+by = i+a(mi −1) =
bni − a, for some non-negative integers x and y. Then a(x + 1) = b(ni − y),
so ni − y is positive. Since a and b are relatively prime, a divides ni − y.
However, ni ≤ a − 1 ⇒ ni − y ≤ a − 1, contradiction.
Therefore, the greatest non-grapefruit is of the form bni − a, ni ≤ a − 1.
The above argument also shows that all positive integers of this form are also
non-grapefruits. Hence, the greatest non-grapefruit is b(a−1)−a = ab−a−b,
proving (1).
43
Now, note that there are mi non-grapefruits in column i. The above tells
us the first grapefruit appearing in column i is ni b. Since 0, b, 2b, . . . , (a−1)b
appear in different columns (because a and b are relatively prime), and there
are a columns, we conclude that as i varies from 0 to a − 1, ni takes on 0, 1,
. . . , a − 1, each exactly once. Therefore, summing over i, 0 ≤ i ≤ a − 1,
X
(i + ami ) =
X
=
X
i
i
i
⇒ a
X
⇒
X
X
ami =
i
bni =
X
a(a − 1)
+a
mi
2
i
a(a − 1)b
2
mi =
a(a − 1)(b − 1)
2
mi =
(a − 1)(b − 1)
,
2
i
i
i+
proving (2).
Finally, suppose that ax1 + by1 = t, and ax2 + by2 = ab − a − b − t, for
some non-negative integers x1 , x2 , y1 , and y2 . Then a(x1 + x2 ) + b(y1 + y2 ) =
ab − a − b, contradicting (1). So, if we consider the pairs (t, ab − a − b − t),
0 ≤ t ≤ (a − 1)(b − 1)/2 − 1, at most one element in each pair can be written
in the given form.
However, we have shown that exactly (a − 1)(b − 1)/2 integers cannot be
written in the given form, which is the number of pairs. Therefore, exactly
one element of each pair can be written in the given form, proving (3).
Remark. There is a much shorter proof using Corollary 2.4. Can you
find it?
For me, this type of problem epitomizes problem solving in number theory,
and generally mathematics, in many ways. If I merely presented the proof
by itself, it would look artificial and unmotivated. However, by looking at
a specific example, and finding a pattern, we were able to use that pattern
as a springboard and extend it into a full proof. The algebra in the proof
is really nothing more than a translation of observed patterns into formal
notation. (Mathematics could be described as simply the study of pattern.)
Note also that we used nothing more than very elementary results, showing
how powerful basic concepts can be. It may have been messy, but one should
never be afraid to get one’s hands dirty; indeed, the deeper you go, the
44
more you will understand the importance of these concepts and the subtle
relationships between them. By trying to see an idea through to the end,
one can sometimes feel the proof almost working out by itself. The moral of
the story is: A simple idea can go a long way.
For more insights on the postage stamp problem, see Ross Honsberger,
“A Putnam Paper Problem”, Mathematical Gems II.
Problems
1. Let a, b, and c be positive integers, no two of which have a common
divisor greater than 1. Show that 2abc − ab − bc − ca is the largest
integer that cannot be expressed in the form xab + yca + zab, where x,
y, and z are non-negative integers.
(1983 IMO)
References
A. Adler & J. Coury, The Theory of Numbers, Jones and Bartlett
I. Niven & H. Zuckerman, An Introduction to the Theory of Numbers,
John Wiley & Sons
c
c
c
c
First Version October 1995
Second Version January 1996
Third Version April 1999
Fourth Version May 2000
Thanks to Ather Gattami for an improvement to the proof of the Postage
Stamp Problem.
This document was typeset under LATEX, and may be freely distributed
provided the contents are unaltered and this copyright notice is not removed.
Any comments or corrections are always welcomed. It may not be sold for
profit or incorporated in commercial documents without the express permission of the copyright holder. So there.
45